2 Supremum of the Airy

DOI:10.1214/ECP.vVOL-PID ISSN:1083-589X

COMMUNICATIONS in PROBABILITY

Upper tail decay of KPZ models with Brownian initial conditions

Patrik L. Ferrari

Bálint Vet˝ o

Abstract

In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [6]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained by Meerson and Schmidt in [18].

Keywords:tba.

AMS MSC 2010:Primary tba, Secondary tba.

Submitted to ECP on tba, final version accepted on tba.

SupersedesarXiv:tba.

1 Introduction

The Kardar–Parisi–Zhang (KPZ) universality class of stochastic growth models in one dimension are described by a stochastically growing interface parameterized by a height function. For general initial conditions, the one-point distribution of the large time limit can be written in terms of a variational problem. The ingredients are the (scaled) initial condition and the Airy2 process,A2. The latter arises as the limiting interface process when the macroscopic geometry of the interface in the law of large numbers is curved. It was discovered in the work of Prähofer and Spohn [23] and described by its finite-dimensional distribution. Soon after, Johansson showed weak convergence of the discrete polynuclear growth model to the Airy2process [16]. In the same paper, a first variational formula appeared (see Corollary 1.3 of [16])

F_GOE(2^2/3s) =P

sup

t∈R

(A₂(t)−t²)≤s

(1.1) whereA2is the Airy2process andFGOEis the GOE Tracy–Widom distribution function discovered in random matrix theory [27]. Formula (1.1) corresponds to the flat initial condition asFGOEis the limiting distribution of the corresponding rescaled interface.

Later, variational formulas describing the one-point distributions for some special initial conditions appeared in several papers, see for instance [3, 24–26]. The first study of a large class of initial conditions, including random initial conditions, is the paper of Corwin, Liu and Wang [8]. In a last passage percolation model they showed the

*Institute for Applied Mathematics, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany. E- mail:ferrari@uni-bonn.de

†Department of Stochastics, Budapest University of Technology and Economics; MTA – BME Stochastics Research Group, Egry J. u. 1, 1111 Budapest, Hungary. E-mail:vetob@math.bme.hu

convergence of the one-point distribution to a probability distribution expressed by the variational formula

P

sup

t∈R

h0(t) +A2(t)−t² ≤s

(1.2) where h0 is the scaling limit of the initial height profile. Shortly after, Remenik and Quastel in [26] asked and answered the question how much discrepancy from the perfectly flat initial condition would be allowed to still see the GOE Tracy–Widom distribution for the KPZ equation. In their paper the variational representation plays an important role. The variational formula approach is proved to be useful since it allows one to go beyond the use of exact formulas and to show, for instance, universal limiting distribution for a flat but tilted profile [11].

Building on [8], Chhita, Ferrari and Spohn derived a variational formula which describes the limiting distribution for random initial conditions which scale to a Brownian motion with the result [6]

F^(σ)(s) =P

sup

t∈R

n√

2σB(t) +A2(t)−t²o

≤s

(1.3) whereBis a standard two-sided Brownian motion independent of the Airy2processA2. This distribution has two special cases which could be analyzed using exact formulas, namelyσ= 0is the flat case and it reduces to (1.1) whereasσ= 1corresponds to the stationary initial condition for the model, so thatF⁽¹⁾(s)is the Baik–Rains distribution [4].

The characterization through a variational formula is tightly related to the question of universality. In the framework of this paper, the key universal ingredient is the Airy2

process which is a projection of a more general space-time random process. The study of this process started with the discovery of the KPZ fixed point by Matetski, Quastel and Remenik [17], for further properties see [5, 20, 21], and continued with the desription of the full space-time process called the Airy sheet or also directed landscape by Dauvergne, Ortmann and Virág [10], see also [19].

Deducing concrete information from a variational formula is however not always an easy task. For example, given (1.3), it is not clear what are the tails of the distribution.

They have only been known for a long time in the cases σ = 0 andσ = 1, because these distributions had other representations, see e.g. [22]. Meerson and Schmidt considered theF^(σ)distribution in [18] and they deduced the correct right tail behavior by a physically motivated but non-rigorous method. They found thatln(1−F^(σ)(s))∼

−⁴₃^√_1+3σ¹ ₄s^3/2fors1. They also performed large scale simulations on the exclusion process confirming their finding.

In this paper we give a rigorous proof of the asymptotics and extend the results of [18] by obtaining upper and lower bound on the prefactor in front of the stretched exponential decay, see Theorem 1.1. The upper tail distribution is governed by the maximal value of√

2σB(t)−t². This fact holds already for non-random initial conditions.

For instance, in the case of (1.1), the tail behaviour of1−FGOE(2^2/3s)matches that of1−FGUE(s)up to the exponential scale as the maximal value of −t²is obtained at t= 0. The same was shown for another simple functionh0in (1.2) as noticed in [28].

To make this point explicit, Theorem 1.3 gives the tail decay for a generic non-random initial condition. One important ingredient for the proof of Theorems 1.1 and 1.3 is the observation that, for allδ >0, the tail distribution of

P

sup

t∈R

A2(t)−δt²

> s

(1.4) is, in the exponential scale, independent ofδ, see Theorem 1.2 for a detailed statement.

For a given realization of the Brownian motion, if the supremum in (1.1) is taken over a finite interval instead ofR, the distribution we are considering also has a Fredholm determinant expression with a kernel depending on the Brownian motion [9]. This representation is however not directly applicable when taking the limit as the finite interval approachesR. For this purpose, it is better to use the kernel given in terms of hitting times noticed first in [26]. That representation works well provided that the functionh₀ is lower than a parabola with a prefactor3/4, while in our application we need to get close to1. The explicit kernel representation has some intrinsic technical issues as confirmed also in the simplest case of a hat-shapedh0 [28]. Our method is mainly probabilistic and avoids the computation of a correlation kernel and its asymptotic analysis.

The second issue that we had to deal with was that the density of the maximum of a Brownian motion with parabolic drift studied first by Groeneboom [13], see also [14]

for explicit formulas, contains a term with a linear combination of AiryAiand AiryBi functions. The leading term is however coming from a subtle cancellation and it does not follow from the naive asymptotic of the Airy functions. Fortunately, we could avoid this issue by using an integral representation discovered by Janson, Louchard and Martin-Löf in [15], which we carefully analyzed asymptotically, see Proposition 3.1.

The paper is organized as follows. We state the main results in the rest of the introduction. We first prove Theorem 1.2 on the upper tail of the supremum of the Airy₂ process minus a parabola with arbitrary coefficient in Section 2. Then Section 3 is about the asymptotic of the supremum of the Brownian motion minus a parabola. Section 4 proves Theorem 1.1 on the right tail of the limiting distributionF^(σ)for Brownian initial conditions. The proof of Theorem 1.3 about the case of general deterministic initial conditions is given in Section 5.

Main results

Theorem 1.1.Letσ >0be fixed. Forslarge enough, the right tail ofF^(σ)(s)satisfies C1s^−3/4e⁻

4 3√ 1

1+3σ4s^3/2

≤1−F^(σ)(s)≤C2s^3/4ln(s)e⁻

4 3√ 1

1+3σ4s^3/2

(1.5) for some constantsC₁, C₂independent ofs.

Thes^−3/4behavior of the prefactor of the lower bound seems to be the correct one.

Indeed, forσ= 0the prefactor iss^−3/4/(4√

2π)(see (1.8) below) and forσ= 1it is given by¹ s^−3/4/√

π. In Proposition 4.1 and 4.2 we give expressions of theσ-dependence of C1, C2.

As it could be expected by the definition of theF^(σ)distribution (1.3), the upper tail behaviour of theF^(σ)distribution is related to tail of the maximum of the Airy2process minus a parabola. The variational formula (1.1) was proved in [16]. In the next result we generalize [16] in the sense that we compute the tail decay for the supremum with parabola which can have any coefficient between0and1.

Theorem 1.2.Let(A2(t))t∈Rdenote the Airy2process. There is a constantC >0such that for allc∈(0,1)

1−FGUE(s)≤P

sup

t∈R

A2(t)−(1−c)t²

> s

≤Cln(s/(1−c)) s^3/4√

1−c e^−43s3/2 (1.6) holds ass→ ∞where the constantCis independent ofs.

1Forσ = 0, the prefactor is obtained from equations (1) and (25), (26) of [2], except that in (26) there is a typo, namelyx^−3/2should bex^−3/4. It can be also easily obtained from the Fredholm determinant representation ofFGOEin [12]. Forσ= 1the distribution is the Baik–Rains distribution, given in Definition 2 of [4]. The prefactor easily follows using (2.3), (2.6) of [4], as well as (26) of [2].

This result might be compared to what has been proven for the KPZ equation with narrow wedge initial condition at finite but large time. In that case, it is known that the decay is exponential withs^3/2power, but without a more precise information on the coefficient, see Proposition 4.2 of [7].

The lower bound in (1.6) is obvious by taking t = 0instead of the supremum. Its asymptotic expansion follows from (1) and (25) of [2], namely

1−F_GUE(s) = 1

16πs^3/2e^−43s3/2(1 +O(s^−3/2)) (1.7) ass→ ∞. We remark that forc≤0the upper bound in (1.6) is trivial as well, since

P

sup

t∈R

A2(t)−(1−c)t²

> s

≤P

sup

t∈R

A2(t)−t²

> s

∼ 1 4√

2πs^3/4e^−43s3/2 (1.8) where we used (1.1) and thex→ ∞tail asymptotic

1−FGOE(x)∼ e^−23x3/2 4√

πx^3/4. (1.9)

Theorem 1.3.Leth₀:R→Rbe a function satisfyingh₀(t)≤A+ (1−ε)t²for allt∈R^, for some constantsA∈R^andε >0. Letκ(h0) = sup_t∈R{h0(t)−t²}and letM >0be large enough so thath0(t)≤κ(h0) + (1−^ε₂)t² for all|t| ≥M. Then there are positive real constantsC1andC2which do not depend on the functionh0ands, such that fors large enough

C₁e^{−43(s−κ(h0))3/2} (s−κ(h0))^3/2 ≤P

sup

t∈R

h₀(t) +A2(t)−t² ≥s

≤C₂Me^{−43(s−κ(h0))3/2}

(s−κ(h0))^1/4. (1.10)

2 Supremum of the Airy

process minus a parabola

The aim of this section is to prove Theorem 1.2 about the upper tail behaviour of the Airy2process minus a parabola with arbitrary coefficient. The first ingredient is a simple bound on the supremum of the Airy2process over a finite interval.

Lemma 2.1.There is a explicit constantCsuch that for alla >0 P

sup

t∈[0,a]

A₂(t)> s

≤Ce^{−43(s−a2)3/2}

(s−a²)^{3/4 (2.1)}

holds ifsis large enough. From this we get the bound

P

sup

t∈[0,a]

A2(t)> s

≤C⁰ a

s^1/4e^−43s3/2 (2.2) for largeswith some other constantC⁰.

Proof. The probability on the left-hand side of (2.1) can be upper bounded as P

sup

t∈[0,a]

A2(t)> s

≤P

sup

t∈[0,a]

A2(t)−t²

> s−a²

≤P

sup

t∈R

A2(t)−t²

> s−a²

= 1−FGOE

2^2/3 s−a²

(2.3)

where we extended the range of tin the second inequality and used (1.1) in the last equality above which is the result of [16]. The proof of the first inequality (2.1) with

C= 1/(4√

2π)is completed by applying the upper tail behaviour of theF_GOEdistribution (1.9).

For (2.2), we divide the interval[0, a] into pieces of length1/√

sand by the union bound we get

P

sup

t∈[0,a]

A2(t)> s

≤

a√ s

X

k=1

P

sup

t∈[(k−1)/√ s,k/√

s]

A2(t)> s

≤C a√

se^{−43(s−1/s)3/2} (s−1/s)^3/4

≤C⁰ a

s^1/4e^−43s3/2

(2.4)

for some constantC⁰ where we applied stationarity of the Airy₂ process and (2.1) in the second inequality and the bound−(s−1/s)^3/2≤ −s^3/2+ 2for alls≥1in the third inequality above.

With this lemma we are ready to prove Theorem 1.2.

Proof of Theorem 1.2. To bound the probability that the Airy2process remains below a parabola, consider an increasing sequencex0= 0< x1 < x2 < . . . to be a partition of R+to be specified later. Then we have by symmetry and the union bound

P

sup

t∈R

A2(t)−(1−c)t²

> s

≤2P ∪^∞_k=0

∃t∈[xk, xk+1] :A2(t)> s+ (1−c)t²

≤2

∞

X

k=0

P ∃t∈[x_k, x_k+1] :A2(t)> s+ (1−c)t²

≤2

∞

X

k=0

P ∃t∈[xk, xk+1] :A2(t)> s+ (1−c)x²_k

≤2C

∞

X

k=0

e⁻⁴³(^s+(1−c)x2k−(x_k+1−x_k)²)^3/2 (s+ (1−c)x²_k−(xk+1−xk)²)^3/4

(2.5) where we decreased the barrier which A2(t) has to reach in [x_k, x_k+1] in the third inequality, while in the forth inequality we used Lemma 2.1 and the translation invariance of the Airy2process.

Thek= 0term in the sum on the right-hand side of (2.5) ise^{−43(s−x21)3/2}/(s−x²₁)^3/4, hence if we choose x1 = 1/√

s, then it is still bounded by a constant multiplied by e^−43s3/2/s^3/4. The rest of the sequence is chosen so that it satisfies

(xk+1−xk)²= 1

4(1−c)x²_k (2.6)

fork= 1,2, . . . which is the geometric choicexk+1 =γxkwithγ= 1 +

√1−c

2 . With this sequence, the right-hand side of (2.5) can be bounded as

P

sup

t∈R

A2(t)−(1−c)t²

> s

≤2Ce^{−43(s−1s)3/2} (s−¹_s)^3/4 + 2C

∞

X

k=1

e⁻⁴³(^s+3₄^(1−c)x2k)^3/2 s^3/4

≤2C⁰e^−43s3/2

s^3/4 + 2C⁰e^−43s3/2 s^3/4

∞

X

k=1

e⁻⁴³(³₄(1−c)γ^2(k−1)s⁻¹)^3/2 (2.7)

where we used the inequality(a+b)^3/2 ≥a^3/2+b^3/2 in the last step. The sum on the right-hand side of (2.7) can be upper bounded using Lemma 2.2 below withα=γ³and β=√

3(1−c)^3/2/(2s^3/2)as

∞

X

k=1

e⁻

√3(1−c)3/2

2s3/2 γ^3(k−1)

≤ ln

1 +^√2γ3s3/2

3(1−c)^3/2

3 lnγ ≤Celn(s/(1−c))

√1−c ^(2.8)

forslarge enough with someCewhich does not depend onc. In the last inequality above we used that lnγ∼ ¹₂√

1−casc →1. The inequalities (2.7) and (2.8) together prove (1.6).

Lemma 2.2.Letα >1andβ >0. Then

∞

X

k=0

e^−βαk≤ ln(1 +α/β)

lnα ^(2.9)

Proof. We start by bounding the sum by an integral as

∞

X

k=0

e^−βαk≤ Z ∞

−1

dx e^−βαx. (2.10)

The change of variablesw=βα^xgives ^dw_dx =wlnαso that the right-hand side of (2.10) Z ∞

−1

dx e^−βαx = 1 lnα

Z ∞

β/α

dwe^−w w =: 1

lnαE₁(β/α). (2.11) Equation (5.1.20) of [1] provides a bound on the exponential integral functionE1, namely E1(x)≤e^−xln(1 + 1/x)forx >0. Thus

∞

X

k=0

e^−βαk ≤ 1

lnαe^−β/αln(1 +α/β) (2.12)

which gives the claimed bound sinceα, β >0.

3 Supremum of Brownian motion minus a parabola

Proposition 3.1.LetG(x) =P maxt∈R(B(t)−¹₂t²)≥x

whereB(t)is a standard two- sided Brownian motion. Then, asx→ ∞we have

G(x) = 3^−1/2e⁻⁴³

√2

3x^3/2(1 +O(x^−1/4)) (3.1) as well as

− d

dxG(x) =2√ 2 3 e⁻⁴³

√2 3x^3/2√

x(1 +O(x^−1/4)). (3.2) Consequently, for anyc >0the density of the random variablemax_t∈R(B(t)−ct²)satisfies

f_c(x) :=− d dxG

(2c)^1/3x

=4 3

√c e⁻⁴³

√4 3c x^3/2√

x(1 +O(x^−1/4)) (3.3) asx→ ∞.

The proof is given below. The distribution function G(x)is written as a contour integral in Lemma 3.5 of [15] as follows:

G(x) = 1 2i

Z

γ

dzHi(z)

Ai(z)Ai(z+ 2^1/3x) (3.4)

whereγis a path passing to the right of all zeroes of the Airy functionAifrom−i∞to i∞. The functionHiis defined by (see (10.4.44) of [1])

Hi(z) =π⁻¹ Z ∞

0

dt e^−t3/3+zt. (3.5)

In [15], the contourγin (3.4) was chosen to come frome^−iθ∞and arrive toe^iθ∞withθ slightly larger thanπ/2. The reason why this contour can be deformed to the vertical one is the following. Lemma A.1 of [15] was used to argue for convergence of the integral (3.4): they showed the decay of the ratio of the two AiryAifunctions, together with a bound onHifor contours with angle more thanπ/2. The bound in Lemma 3.3 is enough to get the convergence also for vertical contours.

For the proof of (3.1), the asymptotic of the Airy andHifunctions will be needed. By (10.4.90) of [1], forxreal,

Hi(x)∼π^−1/2x^−1/4e^23x3/2 asx→ ∞. (3.6) For our purposes, we need also the asymptotic behavior ofHi(z)for complex-valuedz close to2^1/3x/3which we state below and prove later in this section.

Lemma 3.2.Letzbe such that|arg(z)|< π/3. Then for largezwe have the asymptotic behavior

Hi(z)e^−23z3/2 =π^−1/2z^−1/4+O(|z|^−1/2). (3.7) Lemma 3.3.Letθ∈[π/2,3π/2]andx∈R. Then for ally≥0,

Hi x+e^iθy

≤Hi(x). (3.8)

Proof. By the defintion (3.5),

Hi x+e^iθy

=π⁻¹ Z ∞

0

dt e^−t3/3e^xte^eiθyt (3.9) where|e^eiθyt| ≤1for allθ∈[π/2,3π/2]andt≥0. Sincee^−t3/3e^xtis positive, the absolute value of the integral in (3.9) can be upper bounded by the integral ofe^−t3/3e^xt which yields (3.8).

Proof of Proposition 3.1. Let us first prove (3.1). In order to estimateG(x)for largex, we use the integral representation (3.4). The integration contour is chosen to be vertical γ=^21/3₃ x+ iR^{. If}x >0, then forz∈γ, we havearg(z)∈[−π/2, π/2]. Hence we can use the asypmtotics of the Airy function

Ai(z) = ¹₂π^−1/2z^−1/4e^−23z3/2(1 +O(1/z))for|arg(z)|< π, (3.10) see (10.4.59) of [1]. Let us parameterize the pathγasz=^21/3₃ x+ i2^1/3xv,v∈R^.

Contribution for |v| >1/3. Using Lemma 3.3 and 3.2 for theHi function and the asymptotic (3.10) on the Airy functions, the contribution for|v|>1/3is bounded by

Cx^3/4 Z ∞

1/3

dv e^−x3/2g(v), g(v) = 2√ 2 9√

3Re[(4 + 3iv)^3/2−(1 + 3iv)^3/2−1] (3.11) for some constantC. Notice that fora≥0,

d Re((a+ iv)^3/2)

da =−3

2Im√

a+ iv (3.12)

which is an increasing function ofa. This implies thatg(v)is monotone increasing in v ≥ 0 with g(v) ∼√

v asv → ∞. We get that the leading behavior of the integral is bounded byC⁰e^{−x3/2g(1/3)}withg(1/3)−g(0)<0, for some other constantC⁰. Thus this contribution is vanishing with respect to the leading one computed below.

Contribution for |v| ≤ 1/3. For the leading contribution, we use the asymptotic expansion of Lemma 3.2 and (3.10), with the result that the contribution ofG(x)coming from the integral over|v| ≤1/3is given by

2^−2/3x Z

|v|≤1/3

dv Hi

2^1/3

3 x+ i2^1/3xv Ai

2^1/3

3 x+ i2^1/3xvAi

4·2^1/3

3 x+ i2^1/3xv

= x^3/4

√π2^3/4 Z

|v|≤1/3

dv e^x3/2h(v)

(4/3 + iv)^1/4(1 +O(1/x^1/4)) (3.13) with h(v) = ⁴₃√

2((1/3 + iv)^3/2−(4/3 + iv)^3/2/2). Further, one notices that Re(h(v)) is strictly increasing for v < 0 and strictly decreasing for v > 0 with a quadratic approximation for smallvgiven by

h(v) =−4√ 2 3√

3 −3√ 3 4√

2v²+O(v³). (3.14)

Then, using standard steep descent analysis, we get (3.1).

The proof of (3.2) is similar. The only difference is that we need to replace the asymptotic expansion ofAi(z+ 2^1/3x)with the one of2^1/3Ai⁰(z+ 2^1/3x). By (10.4.61) of [1], we haveAi⁰(z) =−¹₂π^−1/2z^1/4e^−2z3/2/3(1+O(1/z)). Thus in the asymptotic analysis we need to replacez^−1/4with−2^1/3z^1/4. This gives the claimed result.

By replacingtby(2c)^−2/3tand using Brownian rescaling, P

maxt∈R B(t)−ct²

≥x

=P

maxt∈R B(t)−¹₂t²

≥x(2c)^1/3

=G

(2c)^1/3x

, (3.15) hence (3.3) follows from (3.2) by substitution.

Now we prove the claimed asymptotic expansion ofHi.

Proof of Lemma 3.2. By symmetry with respect to the real axis (z7→z¯), considerzwith arg(z)∈[0, π/3). We parameterizez=re^iθwithr >0andθ∈[0, π/3). We introduce the change of variablest=√

z+uwhich yields

−1

3t³+zt= 2

3z^3/2−1 3u³−√

zu² (3.16)

and we get

Hi(z)e^−2z3/2/3=π⁻¹ Z ∞

−√ z

du e^−u3/3−

√zu². (3.17)

For large values of|z|, the leading contribution comes from a neighborhood of0. Consider the integration contour Γ = Γ₁∨Γ₂ where Γ₁ = {−√

z+ iy,0 ≤ y ≤ Im(√

z)} and Γ₂={x,−Re(√

z)≤x <∞}.

Consider first the contribution on the contourΓ1. Letf(u) = Re(−u³/3−√

zu²). Then we have

f(−√

z) =−2

3Re(z^3/2) =−2

3r^3/2cos(3θ/2)<0 =f(0). (3.18) Settingu=−√

z+ iy=−√

rcos(θ/2)−i√

rsin(θ/2) + iy, we obtain Re(−u³/3−√

zu²) = const−√

rsin(θ)y, (3.19)

which is decreasing iny. Thus the contribution of the integral of (3.17) overΓ₁ can bounded by the maximum of the integrand times the length of the contour, that is, by π⁻¹Im(√

z)e^f(−√z)≤Ce^{−23r3/2cos(3θ/2)}.

Next we focus on the contribution over Γ2. We have, for all u ≥ −√

rcos(θ/2) =

−Re(√

z), the bound Re(−u³/3−√

zu²) =−¹₃u³−√

rcos(θ/2)u²≤ −²₃√

rcos(θ/2)u². (3.20) For anyδ >0, which can be chosen later as a function ofr, the tails of the Gaussian integral gives

π⁻¹ Z

Γ₂\{|u|≤δ}

du e^−u3/3−

√zu²

≤Ce⁻²³

√rcos(θ/2)δ². (3.21)

Next, the local contribution is close to the integral with only the quadratic term. Indeed, using|e^x−1| ≤ |x|e^|x|, we have

π⁻¹ Z

{|u|≤δ}

du e^−u3/3−

√zu²

−π⁻¹ Z

{|u|≤δ}

du e⁻

√zu²

≤π⁻¹ Z

{|u|≤δ}

du e^−u3/3−

√zu²

|u|³

3 =O(δ³). (3.22) Extending the integration in the Gaussian integral toR, we only make a small error, namely

π⁻¹ Z

{|u|≤δ}

du e⁻

√zu²−π⁻¹ Z

R

du e⁻

√zu²

≤ O(e^−δ2

√rcos(θ/2)). (3.23)

Finally, the Gaussian integral can be computed explicitly as π⁻¹

Z

R

du e⁻

√zu² =π^−1/2z^−1/4. (3.24)

Combining all these bounds, we get Hi(z)e^−23z3/2=π^−1/2z^−1/4+O(δ³,√

re^{−23r3/2cos(3θ/2)}, e⁻²³

√rcos(θ/2)δ²

). (3.25) Now, sinceθ∈[0, π/3), we havecos(3θ/2),cos(θ/2)∈[1/√

2,1]. By choosingδ=r^−1/6, we get (3.7).

4 Tail bounds for random initial conditions

In this section we prove Theorem 1.1 about the upper tail decay of the F^(σ)(s) distribution, which follows by combining Propositions 4.1 and 4.2 below.

Fixc∈(0,1)and let τc= arg max√

2σB(t)−ct²

, Mc= max

t∈R

√

2σB(t)−ct²

=√

2σB(τc)−cτ_c² (4.1) be the position and the value of the maximum of the two-sided Brownian motion√

2σB(t) with diffusion coefficient2σ²whereB(t)is a standard one. Note that

M_c= max

t∈R

√

2σB(t)−ct²

= max

u∈R

√

2σB u 2σ²

−cu² 4σ⁴

d

= max

u∈R

B(u)− c 4σ⁴u²

(4.2) where the second equality follows by the change of variablest=u/(2σ²)and the third one by Brownian scaling. As a consequence, the random variableMc has densityf ^c

4σ4(x) wherefc was defined in (3.3).

Lower bound. The idea of the lower bound is to use the inequality sup

t∈R

√

2σB(t) +A2(t)−t² ≥√

2σB(t0) +A2(t0)−t²₀ (4.3) which holds for any choice of t₀ ∈ R. Furthermore, P √

2σB(t₀) +A₂(t₀)−t²₀> s will be the largest, that is, we get the best lower bound if we take a time√ t0 where 2σB(t0) +A2(t0)−t²₀is the largest. As the Airy2process is stationary and independent ofB, it does not make any difference forA2(t0)which time is chosen. Thus the idea is to chooset0to be the random timeτ1which maximizes√

2σB(t)−t².

Proposition 4.1.For allσ >0, there is a constantC1independent ofssuch that 1−F^(σ)(s)≥C1σ²(1 + 3σ⁴)^1/4s^−3/4e⁻

4 3

√ 1

1+3σ4s^3/2

(4.4) holds forsmax{σ⁻⁴, σ⁴}.

Proof. The upper tail of theF^(σ)distribution can be rewritten as 1−F^(σ)(s) =E

P

sup

t∈R

√

2σB(t) +A2(t)−t²

> s τ1

(4.5) by conditioning on the value of the timeτ1. The conditional probability on the right-hand side of (4.5) can be lower bounded by replacing the supremum of√

2σB(t) +A2(t)−t² with its value att=τ₁to get

1−F^(σ)≥E P√

2σB(τ1) +A2(τ1)−τ₁²> s τ1

=E(P(A2(τ1)> s−M1|τ1))

=E(1−FGUE(s−M1))

(4.6)

where the first equality follows by the definition (4.1) ofM1and by rearranging. In the second equality, we used that the Airy2 process is stationary with GUE Tracy–Widom distribution at any position independently ofτ1.

By using Proposition 3.1 about the asymptotic of the density ofM₁and the tail decay of the GUE Tracy–Widom distribution, see (1.7), one gets that the right-hand side of (4.6) can be lower bounded by

E(1−F_GUE(s−M₁))

≥ 4 3

r 1 4σ⁴

1 16π

Z s

0

dm e⁻

4 3

q₄

3 1 4σ4m^3/2

e^{−43(s−m)3/2}

√m

(s−m)^3/2(1 +R(s, m)))

= 1

24πσ² Z 1

0

dµ e⁻⁴³

q4 3 1

4σ4s^3/2µ^3/2

e^{−43s3/2(1−µ)3/2}

õ

(1−µ)^3/2(1 +R(s, sµ))

(4.7)

with the change of variables m = sµ. Here R(s, m) = O((s−m)^−3/2, m^−1/4) and R(s, sµ) =O(s^−3/2(1−µ)^−3/2, s^−1/4µ^−1/4)is meant ass→ ∞withµ∈(0,1).

Letg(µ) =−⁴₃q

4 3

1

4σ⁴µ^3/2−⁴₃(1−µ)^3/2. One can compute that g⁰(µ) = 0forµ=µ0= 3σ⁴

1 + 3σ^{4 (4.8)}

as well as

g⁰⁰(µ)<0for allµ∈[0,1]. (4.9) In particular, Taylor expansion gives g(µ) = g(µ₀)−α(µ−µ₀)² +O((µ−µ₀)³) with g(µ0) =−⁴₃^{√ 1}

1+3σ⁴ andα= (1 + 3σ⁴)^3/2/(6σ⁴).

The main contribution of the integral on the right-hand side of (4.7) comes from the regime|µ−µ₀| ∼ ¹

s^3/4√

g⁰⁰(µ0) =

√3σ² (1+3σ⁴)^3/4

1

s^3/4. We assume thatsmax{σ⁴, σ⁻⁴}which can be written equivalently ass^−1/4σs^1/4. Next we show that the error terms in Rin (4.7) are small in the regime of the main contribution. Ifσ →0as s→ ∞, then µ₀∼3σ⁴by (4.8) and for the regime which we consider|µ−µ₀| ∼

√ 3σ²

s^3/4 =o(σ⁴)holds as long asσs^−1/4. Henceµs→ ∞andR→0in the regime of the main contribution.

If σ → ∞with s → ∞, then 1−µ0 ∼ _3σ¹4 and the width of the regime considered is

∼₃3/4σ¹³s^3/4 =o(σ⁻⁴)provided thatσs^1/4. Furthermore,(1−µ)s→ ∞andR→0in the regime which gives the main contribution. The errorRalso goes to0in the regime above ifσremains bounded away from0and infinity.

In the regime ofµthat we consider, the higher order terms of the expansion are con- trolled by the quadratic term for allsmin{σ⁴, σ⁻⁴}. Thus the quadratic approximation leads to the lower bound

E(1−FGUE(s−M1))≥ 1 24√

πσ²

õ0

(1−µ₀)^3/2√

αs^3/4e^{−g(µ0)s3/2}(1 +O(s^−1/4))

=σ²(1 + 3σ⁴)^1/4 4√

2π s^−3/4e⁻

4 3

√ 1

1+3σ4s^3/2

(1 +O(s^−1/4)).

(4.10)

Upper bound. This strategy for getting the upper bound is different. We noticed that the tail distribution ofsup_t∈R(A2(t)−(1−c)t²)is, in the exponential scale, independent of c provided that c < 1. This implies that the tail distribution will be determined mostly by the tail ofM_c= sup_t∈R(√

2σB(t)−ct²). The proof of the upper bound goes by conditioning on the value ofM_c and bounding√

2σB(t)−ct²byM_cfrom above.

Proposition 4.2.For allσ >0, there is a constantC2independent ofssuch that 1−F^(σ)(s)≤C2σ⁶(1 + 3σ⁴)⁻²s^3/4ln(s)e⁻

4 3

√ 1

1+3σ4s^3/2

(4.11) holds forsmax{σ⁻⁴, σ⁴}.

Proof. To get an upper bound, one can write the event

sup

t∈R

√

2σB(t) +A2(t)−t²

> s

=n

∃t∈R:√

2σB(t)−ct²

+ A2(t)−(1−c)t²

> so

(4.12) for anyc∈(0,1). Since the maximum of the first term on the right-hand side isM_c, it holds for anyt∈R^that√

2σB(t)−ct²≤Mc, and one can bound the upper tail ofF^(σ)as 1−F^(σ)(s)≤P ∃t∈R:Mc+ A2(t)−(1−c)t²

> s

=E

P

sup

t∈R

A₂(t)−(1−c)t²

> s−M_c M_c

_(4.13)

where the last equality follows by conditioning and rearrangement.

Now the upper bound on the right-hand side of (4.13) can be bounded by an integral

using Proposition 3.1 about the density ofM_cand by Theorem 1.2. Hence we get 1−F^(σ)(s)

≤C Z s

0

dm e⁻⁴³

q4 3

c 4σ4m^3/2

e^{−43(s−m)3/2}

√mln((s−m)/(1−c)) (s−m)^3/4√

1−c

1 +O(m^−1/4)

≤C Z 1

0

dµ e⁻

4 3

q4 3

c

4σ4µ^3/2s^3/2

e^{−43(1−µ)3/2s3/2} s^3/4√

µln((s(1−µ))/(1−c)) (1−µ)^3/4√

1−c

×

1 +O 1 (sµ)^1/4

.

(4.14)

As for the lower bound, we need to havesmax{σ⁻⁴, σ⁴}to apply the approximations.

Very similarly to (4.7), one gets that the exponent is maximal forµ=µ0= _c+3σ^3σ44. We also have

−4 3

r4 3

c

4σ⁴µ^3/2−4

3(1−µ)^3/2=−4 3

√c

√c+ 3σ⁴ −(c+ 3σ⁴)^3/2 6σ⁴√

c (µ−µ0)²+O (µ−µ0)³ . (4.15) This gives

1−F^(σ)(s)≤C⁰ σ⁴

√c+ 3σ⁴√ c

ln(s/(1−c))

√1−c e⁻

4 3

√c

√

c+3σ4s^3/2

(1 +O(s^−1/4)) (4.16) for some constantC⁰ which does not depend oncandσ. Finally, since

√c

√c+ 3σ⁴ = 1

√1 + 3σ⁴ − 3σ⁴

2(1 + 3σ⁴)^3/2(1−c) +O((1−c)²), (4.17) we choose1−c= ˜cs^−3/2. With the choice˜c= ¹₄(1 + 3σ⁴)^3/2/σ⁴, together with (4.16) we obtain

1−F^(σ)(s)≤C⁰⁰σ⁶(1 + 3σ⁴)⁻²s^3/4ln(s)e⁻

4 3√ 1

1+3σ4s^3/2

(4.18) for some other constantC⁰⁰independent ofσ, s.

5 Tail bounds for deterministic initial profile

In this section we prove Theorem 1.3 confirming the heuristics that the leading contribution for the right tail decay comes from the position where the functionh₀(t)−t² is maximal.

Proof of Theorem 1.3. Let τ ∈ R be a time such that κ(h0) = sup_t∈R{h0(t)−t²} = h0(τ)−τ². For the lower bound in (1.10) note that

sup

t∈R

h0(t) +A2(t)−t² ≥h0(τ) +A2(τ)−τ²=κ(h0) +A2(τ) (5.1) by the definition of the timeτ. Hence

P

sup

t∈R

h0(t) +A2(t)−t² ≥s

≥P(κ(h0) +A2(τ)≥s) = 1−FGUE(s−κ(h0)). (5.2) This inequality together with the asymptotic (1.7) leads to the lower bound in (1.10).

Now we consider the upper bound. The functionh0(t)−t²is bounded from above byκ(h0) for all timest ∈ Rand it is bounded from above byκ(h0)−^ε₂t² for |t| > M.

Therefore P

sup

t∈R

h₀(t) +A₂(t)−t² ≥s

≤P

sup

|t|≤M

{κ(h0) +A2(t)} ≥s

+P

sup

|t|>M

n

κ(h0) +A2(t)−ε 2t²o

≥s

≤P

sup

|t|≤M

A2(t)≥s−κ(h0)

+P

sup

t∈R

nA2(t)−ε 2t²o

≥s−κ(h0)

.

(5.3)

The first term is bounded using Lemma 2.1. The second term is bounded using Theo- rem 1.2. Altogether we get

P

sup

t∈R

h0(t) +A2(t)−t² ≥s

≤C⁰ 2M

(s−κ(h0))^1/4e^{−43(s−κ(h0))3/2}+C ln[2(s−κ(h0))/ε]

(s−κ(h₀))^3/4p

ε/2e^{−43(s−κ(h0))3/2}. (5.4) Sinceεis fixed, for largesthe second term is smaller than the first one, which completes the proof.

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Acknowledgments. The work of P.L. Ferrari was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Ex- cellence Strategy – GZ 2047/1, projekt-id 390685813 and by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) – Projektnummer 211504053 – SFB 1060. The work of B. Vet˝o was supported by the NKFI (National Research, Develop- ment and Innovation Office) grants PD123994 and FK123962, by the Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the ÚNKP–20–5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.